Multiple choice

Solve the given quadratic equations and answer the following question. Equation A: (x - 4)2 = (-2x)2 + 45 - 22x - R - 9 Equation B: (6t2 + 7t + 5 6 × ( 12 5 t − 18 ) ) = 0 One root of equation A is 8. The largest root of equation A, when multiplied by the largest prime number less than 60, is equal to

  1. 472

  2. 236

  3. 708

  4. 826

  5. Cannot be determined
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A Correct answer
Explanation

Expanding equation A gives 3x^2 - 14x + 20 - R = 0. Since 8 is a root, substitution gives R = 100, and the other root is -10/3. The largest root is therefore 8, and the largest prime below 60 is 59, giving 8 x 59 = 472.

AI explanation

Substitute x = 8 into equation A to get (8 - 4)^2 = (-16)^2 + 45 - 22(8) - R - 9, which simplifies to 16 = 256 + 45 - 176 - R - 9 and yields R = 100. Substituting R = 100 back into the equation gives (x - 4)^2 = 4x^2 - 22x - 64, and expanding brings it to the standard form 3x^2 - 14x - 80 = 0. Factoring this quadratic equation gives (x - 8)(3x + 10) = 0, so the roots are 8 and -10/3, making the largest root 8. The largest prime number less than 60 is 59, and multiplying 8 by 59 gives 472.

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